Mathematics

Mathematics

Problem

Which of the following integrals gives the exact length of a smooth curve traced once by the vector-valued function r(t)\mathbf r(t) for t∈[a,b]t\in[a,b]? a) ∫ab∣r(t)∣ dt\displaystyle \int_{a}^{b} \lvert\mathbf r(t)\rvert\,dt b) ∫ab∣r′(t)∣ dt\displaystyle \int_{a}^{b} \lvert\mathbf r'(t)\rvert\,dt c) ∫abr′(t) dt\displaystyle \int_{a}^{b} \mathbf r'(t)\,dt d) ∣r(b)−r(a)∣\displaystyle \lvert\mathbf r(b)-\mathbf r(a)\rvert

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